The confluent hypergeometric differential equation

has a regular singular point at

and an essential singularity at

. Solutions analytic at

are confluent hypergeometric functions of the first kind (or Kummer functions):

,

where

are Pochhammer symbols defined by

,

,

. For

, the function becomes singular, unless

is an equal or smaller negative integer (

), and it is convenient to define the regularized confluent hypergeometric

, which is an entire function for all values of

,

and

.

The second, linearly independent solutions of the differential equation are confluent hypergeometric functions of the second kind (or Tricomi functions), defined by

, where the generalized hypergeometric function

represents a formal asymptotic series.

If the hypergeometric function with argument

is complex, both the real and imaginary parts are plotted (black and red curves).

For certain choices of the parameters

and

, the hypergeometric functions are related to various transcendental and special functions. Several illustrations are given in the snapshots.